In this paper, we obtain a new system of canonical integral equations for the plane elasticity problem over an exterior circular domain, and give its numerical solution. Coupling with the classical finite element method, it can be used for solving general plane elasticity exterior boundary value problems. This system of highly singular equations is also an exact boundary condition on the artificial boundary. It can be approximated by a series of nonsingular integral boundary conditions.

This paper revises the definition for the unsolvability of inverse algebraic eigenvalue problems almost everywhere (a.e.) given by Shapiro [5], and gives some sufficient and necessary conditions such that the inverse algebraic eigenvalue problems are unsolvable a.e.

The idea and technique used in [7] are applied to the multiplicative inverse eigenvalue problems as well. Some sufficient and necessary conditions that the multiplicative inverse eigenvalue problems be unsolvable almost everywhere are given. The results are similar to those of [7], but the proofs are more complicated.

Let A be an $n\times n$ nonsingular real matrix, which has singular value decomposition $A=U\sum V^T$. Assume A is perturbed to $\tilde{A}$ and $\tilde{A}$ has singular value decomposition $\tilde{A}=\tilde{U}\tilde{\sum}\tilde{V}^T$. It is proved that $\|\tilde{U}\tilde{V}^T-UV^T\|_F\leq \frac{2}{\sigma_n}\|\tilde{A}-A\|_F$, where $\sigma_n$ is the minimum singular value of A; $\|\dot\|_F$ denotes the Frobenius norm and $n$ is the dimension of A.
This inequality is applicable to the computational error estimation of orthogonalization of a matrix, especially in the strapdown inertial navigation system.

A variable-elliptic-vortex method, which is a generalization of the elliptic-vortex method proposed by the author in [1], is presented for the numerical simulation of incompressible flows. The most attractive feature of the new method is that the numerical vortex blobs used in this model like actual vortex blobs can be translated, rotated and deformed in elliptic shape. The new method provides a more reasonable and more accurate approach for flow simulation than the fixed-vortex methods. Numerical examples are presented to demonstrate the performance of the new method.